In mathematics, an asymptotic formula for a quantity (function or expression) depending on natural numbers, or on a variable taking real numbers as values, is a function of natural numbers, or of a real variable, whose values are nearly equal to the values of the former when both are evaluated for the same large values of the variable.
An asymptotic formula for a quantity is a function which is asymptotically equivalent to the former.

More generally, an asymptotic formula is "a statement of equality between two functions which is not a true equality but which means the ratio of the two functions approaches 1 as the variable approaches some value, usually infinity".

Definition

Let P(n) be a quantity or function depending on n which is a natural number. A function F(n) of n is an asymptotic formula for P(n) if P(n) is asymptotically equivalent toF(n), that is, if


This is symbolically denoted by

Prime number theorem

For a real number x, let π (x) denote the number of prime numbers less than or equal to x. The classical prime number theorem

Prime number theorem

In number theory, the prime number theorem describes the asymptotic distribution of the prime numbers. The prime number theorem gives a general description of how the primes are distributed amongst the positive integers....

 gives an asymptotic formula for π (x):

Stirling's formula

Stirling's approximation formula

Stirling's approximation

In mathematics, Stirling's approximation is an approximation for large factorials. It is named after James Stirling.The formula as typically used in applications is\ln n! = n\ln n - n +O\...

 is a well known asymptotic formula for the following quantity:
.

The asymptotic formula is

Asymptotic formula for the partition function

For a positive integer n, the partition function P(n), sometimes also denoted p(n), gives the number of ways of writing the integer n as a sum of positive integers, where the order of addends is not considered significant. Thus, for example, P(4) = 5. G.H. Hardy and Srinivasa Ramanujan

Srinivasa Ramanujan

Srīnivāsa Aiyangār Rāmānujan FRS, better known as Srinivasa Iyengar Ramanujan was a Indian mathematician and autodidact who, with almost no formal training in pure mathematics, made extraordinary contributions to mathematical analysis, number theory, infinite series and continued fractions...

 in 1918 obtained the following asymptotic formula for P(n):

Asymptotic formula for Airy function

The Airy function

Airy function

In the physical sciences, the Airy function Ai is a special function named after the British astronomer George Biddell Airy...

Ai(x) which is a solution of the differential equation


and which has many applications in physics, has the following asymptotic formula: